TRAJECTORY TRACKING CONTROLOFAN … · 2014. 9. 20. · Abstract— This paper considers the...

TRAJECTORY TRACKING CONTROL OF AN OMNIDIRECTIONAL WHEELED

MOBILE ROBOT WITH SLIP AND DEFORMATION: A SINGULAR

PERTURBATION APPROACH

C. A. Pena Fernandez.∗ Jes J. F. Cerqueira∗ Antonio M. N. Lima†

∗Robotics Laboratory - Department of Electrical Engineering, Polytechnic School,

Federal University of Bahia

Rua Aristides Novis, 02, Federacao, 40210-630, Salvador, Bahia, Brasil

Telefone:+55-71-3203-9760.

†Department of Electrical Engineering at Center of Electrical and Computer Engineering,

Federal University of Campina Grande

Rua Aprigio Veloso, 882, Universitario, 58429-970, Campina Grande, Paraıba, Brasil

Telefone:+55-83-2101-1000.

Email: [email protected],[email protected],[email protected]

Abstract— This paper considers the trajectory tracking control of a omnidirectional wheeled mobile robot(OWMR) with slip and deformation in the wheels, i.e., when the kinematic constraints are not satisfied. Thedynamic of the OWMR is give in formalism with the aid of the Lagrange approach and the singular perturbationstheory. The proposed controller guarantees that the tracking error converges to small ball of the origin by choosingsmall values of slip (<0.1) and by selecting appropriate parameters of an auxiliary control law. To design thecontroller, it is used a integral manifold to linearize the dynamic model of the OWMR and consequently controlit. To this end, the principle of Poincare-Lindstedt is used, i.e., the manifold and the control inputs are rewritelike a Taylor expansion. The singular perturbations theory allows to manipulate the flexibility through of asmall factor in the dynamic model (normally, known as ε) at the same time that scales the dissatisfaction of thekinematics constraints. Thus, we will observe the behavior of the tracking when the controller is applied to suchmodel.

Keywords— Omnidirectional mobile base, slip, flexibility, tracking trajectory, singular perturbations.

Resumo— Este artigo aborda o problema de controle de seguimento de trajetoria de um robo movel omni-direcional (RMO) com deslizamento e deformacao nas rodas, ou seja, quando as restricoes cinematicas nao saosatisfeitas. O modelo dinamico do RMO e dado por um formalismo baseado em uma abordagem Lagrageana e deperturbacoes singulares. O controlador proposto garante com que o erro de seguimento convirja a uma vizinhancada origem pela escolha de pequenos valores de deslizamento (<0.1) e pela selecao apropriada dos parametros emuma lei de controle auxiliar. Para a projecao desse controlador e usado uma variedade (manifold em ingles) paralinearizar o modelo do RMO e consequentemente controla-lo. Para esse fim, o princıpio de Poincare-Lindstedte usado, ou seja, a variedade e as entradas de controle sao reescritas como uma expansao de Taylor. A teoriade perturbacoes singulares permite nao so manipular a flexibilidade atraves de um pequeno fator no modelodinamico (usualmente, conhecido como ε), mas tambem escala a insatisfacao das restricoes cinematicas. Dessaforma, sera observado o comportamento do controlador de seguimento de trajetoria proposto quando este sejaaplicado no modelo dinamico.

Palavras-chave— Base movel omnidirecional, dinamica do escorregamento, seguimento de trajetoria, pertur-bacoes singulares.

1 Introduction

The design of feedback-control laws for mechanicalsystems subjected to kinematic constraints has be-come an area of great interest (Fernandez et al., 2013;Bazzi et al., 2014). This is the case of the stabiliza-tion and tracking problems of omnidirectional wheeledmobile robots (OWMRs). The stabilization has beenan extensive research area in past decades due to itschallenging theoretical nature, i.e., an intrinsic non-linear control problem and its practical importance. It is well-known that there does not exist a smo-oth pure state feedback control law1 such that thestate of a wheeled mobile robot converges to the ori-gin (Fernandez et al., 2013). In order to mitigate thisdifficulty, several types of controllers have been propo-sed, such as time-varying control laws, discontinuouscontrol laws, and hybrid control laws (For more de-

1Consequence of the Theorem 1, pp. 186 in Brockett(1983).

tails, see (Kolmanovsky and McClamroch, 1995)).

The tracking problem of WMRs has also been stu-died. The techniques for trajectory control has beenbased on linearization techniques for local controlling(Walsh et al., 1994); in techniques of nonlinear statefeedback with singular parameters (Motte and Cam-pion, 2000; Leroquais and D’Andrea-Novel, 1996);or also in techniques based on backstepping (Jiang,2000). A condition usually considered for trajectorytracking problems in OWMRs is the ideal rolling as-sumption, i.e., the wheels are assumed to roll withoutslip, or equivalently, with kinematic constraints sa-tisfied. In principle, the slip is associated to variouseffects such as deformability or flexibility of the whe-els (Leroquais and D’Andrea-Novel, 1996; Fernandezet al., 2012; Fernandez and Cerqueira, 2009b; Fer-nandez and Cerqueira, 2009a). Thus, disregardingthe slip of the wheels in the dynamic model leadsus to path tracking problems (Fernandez and Cer-queira, 2009a; Motte and Campion, 2000). If the slip

Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014

32

(a)

L

x1

x2 m, Ic

θ

ϕ1, Iw

ϕ2, Iw

ϕ3, Iw

p

{G}

{L}

(b)

Figure 1: OWMR with three omnidirectional wheels. a) OmniBot b) Configuration of the OWMR.

is not taken into account, a designed task may notbe completed and a stable system may even becomeunstable (Terry and Minor, 2008). In this paper, theslip and deformation of the wheels have been conside-red into the dynamic model of the OWMR such thatit can be modeled by a singularly perturbed model.We will study the trajectory tracking control of theOWMRs subject to these two effects and aim at de-signing a robust controller based in a nonlinear statefeedback. To this end, with aid of the inverse dyna-mic a robust controller is designed to make a nonlinearstate feedback. On the other hand, by assuming thatthere exists an invariant and integrable manifold forthe kinematic constraints (known as fast variables inthe singular perturbation approach), corrective con-trol actions can be projected and added to the con-troller such that slip in the wheels can be mitigated(Fernandez et al., 2012; Fernandez et al., 2013). Weapply our methodology to validate a control law withcorrective actions for the trajectory tracking problemsin the OmniBot model, an OWMR that is being cons-tructed in the Robotics Laboratory at University Fe-deral of Bahia for research and development of thetrajectory controllers (see Fig. 1(a)).

This paper is organized as follow: in Section 2is showed the mathematical model and the prelimi-naries foundations associated with inclusion of slip inthe singularly perturbed model. In Section 3 an in-variant manifold for the kinematics constraints (asso-ciated with fast variables) is defined and computedsuch that corrective control actions can be projectedto minimize the errors into the control scheme due tothe slip. In Section 4 the proposed approach is usedfor accessing the achievable closed-loop performance.Finally, conclusions and closing remarks are shown inSection 5.

2 Dynamic model and foundations

This paper consider the configuration of a OWMRwith three omnidirectional wheels, as shown in Figure1(a). Such configuration can be fully described by thevector q ∈ R

6 of generalized coordinates defined by

q =[

x1 x2 θ ϕ1 ϕ2 ϕ3

]T

where {x1, x2, θ} is the set of coordinates associatedwith the cartesian position of the body frame {L} into

the global frame {G} and guidance of mobile base, theset {ϕ1, ϕ2, ϕ3} is associated with the angular positionof each wheel (which can not be controlled indepen-dently) (see Fig. 1(b)).

The kinematic constraints can be expressed like aPfaffian constraint (Motte and Campion, 2000):

AT (q)q = 0 (1)

where A(q) is the matrix with the holonomic kinema-tic constraints defined by

A(q) =

− sinα sin θ sinβ− cosα − cos θ − cos β

b b br 0 00 r 00 0 r

,

[

A1(θ)

A2(θ)

]

, A(θ),

where α = θ − 2π3, β = θ + 2π

3, b is the displacement

from each of driving wheels to the axis of symmetryof the OWMR and r is the radius of each wheel.

Provided that the ideal kinematic constraints arenot satisfied [i.e., AT (q)q 6= 0] then the generalizedvelocity vector q may be written as

q = S(q)v + A(q)εµ (2)

being µ = [ µ1 µ2 µ3 ]T an instrumental vector in senseof accessing the violations of the ideal kinematic cons-traints in the OWMR (Fernandez et al., 2013), v =[ vn ω ]T is the vector that contains the linear (vn) andangular (ω) velocities and

S(q) =

cos θ sin θ 0− sin θ cos θ 0

0 0 1−√

3/2r 1/2r b/r0 −1/r b/r

√3/2r 1/2r b/r

,

[

S1(θ)

S2(θ)

]

, S(θ),

is the Jacobian. The term ε is a scale factor associatedwith the flexibility of the dynamic model (Fernandezet al., 2013).

Property 1 The matrices A(q) and S(q) satisfyAT (q)S(q) = 0 (D’Andrea-Novel et al., 1995).

As usual, the dynamic model for such mobile baseis given by

Mq = Λ +Buε + A(θ)λ (3)

where M = diag (m m Ic Iw Iw Iw ), Λ = 06×1, B =[

03×3

I3×3

]

are the inertia matrix, the centripetal and co-

riolis torques (It is assumed that the geometrical cen-ter coincides with the mass center, thus this vector


33

−15 −10 −5 0 5 10 15−2000

−1500

−1000

−500

0

500

1000

1500

2000

s

Fy

(a) By=0.178,Cy=1.65,Dy=2193,Ey=0.432

−15 −10 −5 0 5 10 15−2500

−2000

−1500

−1000

−500

0

500

1000

1500

2000

2500

δx

Fx

(b) Bx=0.244,Cx=1.30,Dx=1936,Ex=-0.132

Figure 2: Graphical representation of Fx and Fy. In both cases FN = 2000N .

is equal to zero) and a full rank matrix, respectively.The parameter m is the mass of the WMR, Ic is themoment of inertia of the WMR on a vertical axis th-rough the intersection of the axis of symmetry withthe driving wheel axis and Iw is the moment of inertiaof each driving wheel on its axis. The vector uε re-presents the input torques provided by the actuatorsand λ ∈ R

3 represents the Lagrange multipliers (Blochet al., 2000).

2.1 Traction forces

The traction forces, the transversal force Fx and thelongitudinal force Fy, are represented in terms of theslip angle δx and the longitudinal slip ratio of eachwheel, si, respectively. Bakker et al. (1987) propo-ses the following functions to represent these tractionforces:

Fy = Dy sin(

Cy tan−1

(

By

[∥

∥

∥(1− Ey)100s

+

(

Ey

Bytan−1(100Bys)

)]))

(4)

Fx = Dx sin(

Cx tan−1 (Bx [(1− Ex)δx

+

(

Ex

Bxtan−1(Bxδx)

)]))

(5)

being Bx, Cx, Dx, Ex, By , Cy , Dy and Ey defined by

Cx = 1.30 , Cy = 1.65

Dx = −α1F2N − α2FN , Dy = −Dx

Bx =α3 sin

(

α4 tan−1(α5FN )

)

CxDX

By =α3F

2N + α4FN

CxDxeα5FN

Ex = Ey = α6F2N + α7FN + α8

where α1, . . . , α8 are known constants and FN is a nor-mal force to surface. In order to linearize the equations(4) and (5) will be used the following assumption:

Assumption 1 ((Motte and Campion, 2000))The slip and slip angle are limited signals

−∆y ≤ s ≤ ∆y and −∆x ≤ δx ≤ ∆x

for |∆y| = 0.1 and |∆x| = 0.1 such that the tractionforces Fx, Fy belong to the linear region of the equati-ons (5) and (4).

By using the Assumption 1, the functions sin(·)and tan−1(·) may be replaced by the linear function(or identity)2 and the equations (5) and (4) can berewritten as

Fx = DxCxBxδx = D δx (6)

Fy = 100DyCyBys = Gs (7)

where D and G are the stiffness coefficients for thetransversal and longitudinal movements of each wheel,respectively.

2.2 Singularly perturbed model

In practice, the constraint (1) does not hold. Multi-plying both sides of (2) by AT (q), and by using theProperty 1 is obtained that

AT (q)q = AT (q)A(q)εµ. (8)

Remark 1 In OWMRs the matrix AT (q) is associa-ted with the equations that restrict the motions withslip. Phenomenologically, these equations representthe speed of the contact point of the wheel.

If ε = 0 then (8) becomes the ideal constraint(1). In other words, the parameter ε governs the dis-satisfaction of the kinematic constraints and it mustbe included into the dynamic model. To this end, wepropose a singularly perturbed dynamic model for theWMR, like in (Fernandez et al., 2013), defined by thefollowing state-space:

x = B0(q)v + [εB1(q) +B2(q)]µ+B3(q)uε

εµ = C0(q)v + [εC1(q) +C2(q)]µ+ C3(q)uε

y = P0(q)

(9)

(10)

(11)

where x = [ qT vT ]T can be used to denote the “slow”variables and µ beyond its instrumental meaning canbe used to denote the“fast”variables; uε = [ uε,1 uε,2 ]T

2Since the results are expressed in radians.


34

has the manipulated inputs associated with the tor-ques at the motors and y = [ y1 y2 ]T has the cartesiancoordinates of a point p located at a distance L of thesymmetry axle of the WMR, i.e., we define:

y = [ y1y2 ] = P0(q) ,[

x1−L sin θx2+L cos θ

]

. (12)

The matrices Bi(q), Ci(q), for i = 0, 1, 2, 3, aresuccessively:

B0(q) =

[

S(θ)a1 a2 a3

a4 a5 a6

a7 a8 a9

]

, B1(q) =

[

A(θ)a10 a11 a12

a13 a14 a15

a16 a17 a18

]

,

B2(q) =

[

06×3

a28 a29 a30

a31 a32 a33

a34 a35 a36

]

, B3(q) =

[

06×3

a19 a20 a21

a22 a23 a24

a25 a26 a27

]

,

C0(q) =[ a37 a38 a39a40 a41 a42

a43 a44 a45

]

, C1(q) =[ a46 a47 a48a49 a50 a51

a52 a53 a54

]

,

C2(q) =[ a64 a65 a66a67 a68 a69

a70 a71 a72

]

, C3(q) =[ a55 a56 a57a58 a59 a60

a61 a62 a63

]

,

being ai, for i = 1, . . . , 72, known values and definedby

ai , ai(V, δ,Do, Go),

where the parameter V is the velocity of the wheelcenter and δ is a “small” positive constant to avoid thenumerical problem for small values of V (i.e., for smallvalues of V , it is replaced by V + δ). The parametersDo and Go are normalized values defined by

Do = εD and Go = εG, (13)

where D and G are the stiffness coefficients defined in(6) and (7).

Assumption 2 The longitudinal and transversalstiffness coefficients (G and D, respectively) are thesame for the three wheels and

ε = inf{1/G, 1/D}.

Assumption 3 The velocities of both driving wheelsat their center are taken to be identical, and more pre-cisely, equal to their average:

V =(

x21 + x2

2 + θ2)1/2

. (14)

Remark 2 When ε = 0 the model defined by (9)-(11)is called rigid model. When ε 6= 0 the model is calledflexible model (D’Andrea-Novel et al., 1995).

2.3 Boundary-layer system

Like in (D’Andrea-Novel et al., 1995), assuming thatthe control input uε is a smooth function of timeuε , uε(q, v) then, for ε = 0, the equation (10) can berewritten as follows:

C0(q)v +C2(q)µ+ C3(q)uε(q, v) = 0. (15)

Definition 1 ((D’Andrea-Novel et al., 1995))The model defined by (9)-(11) is in standard form ifonly if (15) has k ≥ 1 distinct isolated roots.

Indeed, a root of (15), denoted by µ, is

µ = −C−12 (q) [C3(q)uε(q, v) + C0(q)v] , (16)

thus the reduced system associated is obtained bysubstituting (16) in (9):

˙x =B0(q)v − [εB1(q) +B2(q)]C−12 (q) [C3(q)×

uε(q, v) + C0(q)v] +B3(q)uε;

x(0) = x0, (17)

and the boundary layer system is

dµ

dτ=C0(q)v0 + [εC1(q0) + C2(q0)] (µ+ µ)

+ C3(q0)uε;

µ(0) = µ0 − µ0, (18)

where τ = t/ε, v0, q0 are interpreted as fixed para-meters and µ(0) = µ0 − µ0 being µ0 equal to (16)evaluated in v0, q0.

Now, we introduce two conditions:

Condition 1 There exist T, λ1, λ2, ε0 and the ballsZ1 = (0;λ1), Z2 = (0; λ2) such that

• The matrices Bi(q) and Ci(q) in the model (9)-(11) (for i = 0, . . . , 3) and their partial derivati-ves with respect to x, µ and ε are continuous inZ1 × Z2 × [0, ε0]× [0, T ],

• The function (16) and εC1(q) + C2(q) have con-tinuous first partial derivatives,

• The reduced system (17) has an unique solutionx defined on [0, T ] which belongs to Z1.

Condition 2 µ = 0 is an exponentially stable equi-librium point of the boundary layer system (18) uni-formly in the parameter x0. Furthermore, µ0 − µ(0)belongs to its domain of attraction. This condition im-plies that µ(τ ) exists for τ ≥ 0 and that lim

τ→+∞µ(τ ) =

0.

The Tikhonov’s theorem states the relationbetween x and x on one hand and µ, µ and µ on theother hand.

Theorem 1 (Tikhonov’s theorem) For a systemin a standard form, if the Conditions 1 and 2 are sa-tisfied, then there exist positive constants ν1, ν2 and ε∗

such that if ‖x0‖ < ν1, ‖µ0−µ0‖ < ν2 and ε < ε∗ thenthe following approximations are valid for t ∈ [0, T ]:

x(t) =x(t) +O(ε) (19)

µ(t) = µ(t) + µ(τ ) +O(ε) (20)

where O(ε) represents a quantity of the order of ε.

The Theorem 1 implies that there exists t1 > 0such that the approximation

µ(t) = µ(t) +O(ε)

is valid for t ∈ [t1, T ]. Leaving only choose an appro-priate value for ε, such that the Theorem 1 is satisfied.


35

3 Controller design

Like (Fernandez et al., 2013), the controller is designedby using the singular perturbation theory. Thus, aninvariant manifold for the fast variables is introducedand a nonlinear state feedback is imposed in order tolinearize the system.

3.1 Computing the invariant manifold

In order to linearize the dynamic model an invariantmanifold for the fast variables is introduced and defi-ned by

µ = Hε(x, uε, ε). (21)

The purpose of this subsection is to use the ma-nifold in (21) to make that the tracking error will con-verge to zero when a state feedback control law uε isapplied.

By considering the principle of Poincare-Lindstedt, we prefer to construct the linearizing con-trol law as well as the corresponding Hε assumingthese functions to be analytic. Thus, these functionsand their time derivatives can be developed under theform of Taylor series expansion:

uε = u0 + εu1 + ε2u2 + . . .+ εNuN (22)

Hε = H0 + εH1 + ε2H2 + . . .+ εNHN (23)

Hε = H0 + εH1 + ε2H2 + . . .+ εNHN , (24)

where N can be considered a robust term.By substituting (22)-(24) in (9)-(10) gives

x = B0(q)v +B2(q)H0 +B3(q)u0+

ε [B1(q)H0 +B2(q)H1 +B3(q)u1] +

ε2 [B1(q)H1 +B2(q)H2 +B3(q)u2] + . . .

εN [B1(q)HN−1 +B2(q)HN +B3(q)uN ] (25)

and

ε[

H0 + εH1 + ε2H2 + . . .+ εNHN

]

=

C0(q) v + C2(q)H0 + C3(q)u0+

ε [C1(q)H0 + C2(q)H1 + C3(q)u1] +

ε2 [C1(q)H1 +C2(q)H2 + C3(q)u2] + . . .

εN [C1(q)HN−1 + C2(q)HN +C3(q)uN ] , (26)

and equating like powers of ε in (26) gives the followingrecursive expression for the termsHk, for i = 1, . . . , N ,in (23):

Hk = C−12 (q)

[

Hk−1 −C1(q)Hk−1 −C3(q)uk

]

. (27)

For k = 0, (27) do not offer sufficient information tocalculate H0 in (23). So, from (10), for ε = 0, thecomponent H0 can be calculated as

H0 = −C−12 (q) [C0(q) v + C3(q)u0] , (28)

where u0 is the first component into the Taylor se-rie expansion defined in (22). The equation (28) isknowing as “slow” manifold. The terms Hk for k > 0implies that the trajectories of system (9)-(11) moveon a slight variation of the “slow” manifold3. In the

3That slight variation of the “slow” manifold is calledthe “fast” manifold.

same way, uk for k > 0 implies that the control law iscomposed by the main control law, in this work defi-ned by u0, and the corrective control actions (i.e., theterms with powers of ε greater than 0 in (22)).

3.2 Computing the corrective control actions

From (25), the set of corrective controls u1, u2, . . . , uN

are simply designed to annihilate the set of termsε, ε2, . . . , εN , respectively. That is,

u1 = −B+3 (q) [B1(q)H0 +B2(q)H1]

u2 = −B+3 (q) [B1(q)H1 +B2(q)H2]

...

uk = −B+3 (q) [B1(q)Hk−1 +B2(q)Hk] (29)

for k = 1, 2, . . . , N , where B+3 (q) is the pseudo-inverse

of B3(q) (Fernandez et al., 2013).

3.3 Computing the control law u0

The control law u0 = uε(q, v)∣

∣

ε=0is projected by using

the inverse dynamics of (3) and the second derivativeof (2). Thus,

q =

[

∂S

∂qS(q)v

]

v + S(q)v. (30)

Eliminating Lagrange multipliers in (3) and usingthe relation (30) give

v =[

ST (q)MS(q)]−1

ST (q)[∥

∥

∥Bu0−

M

[

∂S

∂qS(q)v

]

v

]

. (31)

Consequently, the law uε is defined by the inverse of(31):

u0 =[

ST (q)B(q)]−1 {∥

∥

∥ST (q)

[∥

∥

∥M(q)S(q)η

+ M(q)

[

∂S

∂qS(q)v

]

v

]}

. (32)

Remark 3 By substituting (32) in (31) is obtainedη = v, where η is an auxiliary control variable.

3.4 Trajectory tracking problem

Essentially, the trajectory tracking problem is to find astate feedback controller that can achieve the tracking,with stability, of a given moving reference position yref:

yref =

[

y1,ref

y2,ref

]

: R+ → R2

which is assumed to be twice differentiable.More precisely, the objetive is to find an auxiliary

state feedback law η such that: i) the tracking errors‖y1−y1,ref‖, ‖y2−y2,ref‖ and the control η are boundedfor ∀ t; ii) the tracking errors are time invariant; andiii) limt→∞ ‖y1 − y1,ref‖ = 0, limt→∞ ‖y2 − y2,ref‖ = 0.

If the reference trajectory is such that yref, yref,yref are bounded for every t, then the trajectory trac-king problem is solvable for any restricted mobilityOWMR, by using the following auxiliary control law:

η = ∆−1(q)[

yref −K2˙y −K1 y − ∆(q) v

]

(33)


36

−200 0 200 400 600 800 1000 1200 1400 1600−200

0

200

400

600

800

1000

1200

1400

1600

−5 0 5

−5

0

5

y1 (m)

y2

(m)

(a) Unstable behavior of the WMR for tracking trajectorywhen ε = 5× 10−5 with control law (32).

10−8

10−7

10−6

10−5

0

500

1000

1500

2000

2500

3000

3500

4000

Nε = 0.0

nε = 5Nε = 0.0

nε = 8

Nε = 0.4

nε = 5

Computationalcost

(s)

ε

(b) Simulation of the computational cost of the control law(32). Imminent instability when ε ≥ 5 × 10−5 (Nε = 0.8and nε = 11).

Figure 3: Simulation of the computational cost of the control law (32) according to the values of ε defined by ε =10−nε +Nε10−nε+1 and tracking trajectory for ε = 5× 10−5.

where y = y − yref is the tracking error, {K1 , K1} ⊆R

2×2 are arbitrary positive definite matrices such thatthe desired dynamic represented by yref + K2 yref +K1 yref is Hurwitz and

∆(q) =∂P0(q)

∂yS1(θ) =

[

cos θ sin θ −L cos θ− sin θ cos θ −L sin θ

]

.

One can to conclude that the control law (33)increases its robustness in that it increases the degreeof the Taylor’s expansions (22), (23) and (24). Fora better understanding, we will expresse the trackingerror and its derivative by

y = y − yref˙y = P0(q)− yref = P ∗

0 (q)− yref

= P ∗0 (S1(θ)v + A1(θ)εHε)− yref

where

P ∗0 (q) =

[

x1−θL cos θ

x2−θL sin θ

]

and Hε is the manifold defined in (21).

4 Evaluating the controller

In order to verify effectiveness of the proposed control-ler, simulations were done by using the robust nonli-near state feedback based controller defined by (32)and (33). However, we must first know the appropri-ate value for ε due that Theorem 1 imposes the limitε∗.

Remark 4 The value of ε is associated with the fle-xibility and deformation of the wheels, we can say thatwhen the value of ε increases then the computationalcost of the control law (32) also increases. This isa direct consequence of the friction coefficient, whichalso increases when the deformation of the wheels in-creases and it is a significant cause of the dead zone inthe actuator. So, the computational effort is associa-ted with an attempt of the control law (32) to overcomethe dead zone (Fernandez and Cerqueira, 2009b).

4.1 Choosing the value of ε∗

Unlike other works (Leroquais and D’Andrea-Novel,1996; Motte and Campion, 2000; D’Andrea-Novelet al., 1995; Fernandez et al., 2013), a value is set forthe maximum parameter perturbation, ε∗. Let consi-dered the following transformation on ε for a betternumerical manipulation:

ε = 10−nε +Nε10−nε+1 (34)

being nε ∈ Z+ and Nε ∈ [0, 1] ⊂ IR+. Assuming thatthe coefficients D and G are the same for all wheelsand by using of (13) and the Assumption 2, we chosenD0 = G0 = 1 N.

We assume that the desired trajectory is a rhom-bus with the diagonals equal to 6.28 m. Each simu-lation represent a duration of 4.5 s. The numericalvalues used in the simulations are: m = 1.903 Kg,Ic = 0.0132 Kg-m2, Iw = 1.6× 10−5 Kg-m2, L = 0.12m, b = 0.12 m and r = 0.0349 m. The auxiliary con-trol law was set with K1 = 70 I2×2 and K2 = 23 I2×2,equivalent with a overdamped second-order systemwith poles at -19.38 and -3.61.

In Fig. 3(b) is shown the computational cost (me-asured in seconds) for the interval [10−9, 5× 10−5] inthe domain of ε. It can be seen that the evolutionof the computational cost increases when ε increases.Particularly, when ε = 5×10−5 (Nε = 0.4 and nε = 5)the system is unstable for the tracking, as shown inFig. 3(a). Thus, it is defined ε∗ = 5 × 10−5 in theTheorem 1. Values greater than ε∗ unstabilize thesystem.

4.2 Dynamic model and Trajectory tracking

In Fig. 4 is shown the evolution of the vector µ anddemonstrated that µ = 0 guarantee the Condition 2.Thus, it is possible to assert that the dynamic modeldefined by (9)-(11) satisfies the approximations (19)


37

−50

0

50

100

150

−200

−150

−100

−50

0

50−10

−5

0

5x 10

−3

µ1 (m/s)µ2 (m/s)

µ3(m

/s)

µ = 0

Figure 4: Evolution of the vector µ, 3D view: µ1× µ2× µ3.

and (20), i.e.,

q(t) = q(t) +O(ε),

v(t) = v(t) +O(ε) e

µ(t) = µ(t) +O(ε).

The above approximations allow to compute the inva-riant manifold Hε into the singular perturbation ap-proach. To observe the behavior of the control law(32) when it is applied in the model defined by (9)-(11), we can study the cases in which the model istotally rigid (ε = 0) and flexible (ε 6= 0), according tothe Remark 2. It was used one corrective action, i.e.,N = 1 in (22) and (23):

uε = u0 + εu1 (35)

and

Hε = H0 + εH1.

In Fig. 5(a) is shown the tracking made by thecontrol law (35) when ε = 4 × 10−11 such that thecondition ε < ε∗ (= 5 × 10−5) in the Theorem 1 issatisfied.

The Fig. 5(a) presents the tracking of the tra-jectory for four different velocities: 2.34 cm/s, 4.68cm/s, 7.02 cm/s and 9.36 cm/s. For each velocity, itcan be observed that the deviation with respect to thefirst corner of the rhombus is incremented with eachvelocity increase. A better detailing about the devi-ations associated with both cases (when ε = 0 andε 6= 0) is presented in Table 1. All these deviationswere measured by taking the maximum distance withrespect to the first corner of the rhombus. It can be no-ted that when the velocity vn increases the deviationsare also larger (see the first column in the Table 1).Possibly, the models of longitudinal and transversaltraction forces, represented by the equations (7) and(6), have insufficient information to model the truebehavior of the slip, thus the controller designed willhave a greater vulnerability to reject the disturbancesproduced by the slip when vn is increased.

By setting ε = 0 the dynamic model becomes ri-gid. In Fig. 5(b) is shown the behavior of the tracking.Clearly, this behavior presents greater deviations thanthe case ε 6= 0 (compare the second column in Table1).

Table 1: Deviations of tracking by the OWMR.

vn [cm/s] Case ε 6= 0 [cm] Case ε = 0 [cm]

2.34 5.564× 10−2

50.40

4.68 1.44 148.17

7.02 4.66 19.64

9.36 14.23 80.89

5 Final remarks

In this paper, the path tracking control problem of aOWMR with slip and deformations has been conside-red. A robust controller based in a nonlinear statefeedback for the dynamic model of the OWMR alsohas been proposed. The dynamic model was consi-dered by using the singular perturbations theory (seeequations (9)-(11)).

In others related work OWMRs (Huang et al.,2012; Stonier et al., 2007; Barreto et al., 2013) havenot been considered the change in the velocity vn inthe projection of the trajectory tracking controller. Inthis work was considered a reference trajectory thatcould explore the critical situations in which the in-crement of velocity vn incurs a significant increase inthe slip. The rhombus conveniently contributes to thisobservation because the corners represent a suddenchange of direction and the OWMR tends to leave ofthe trajectory. In this situation and due to the trac-tion forces are linear (see (6) and (7)), the controllerhas not the ability to calculate the appropriate controlaction uε. As a consequence of this, the controller pro-posed here was designed using corrective actions suchthat the deviations could be reduced.

The controller was designed in two parts: on theone hand, the control law (32) designed by inverse dy-namic (compare (31) and (32)) and the other hand theauxiliary control law defined by (33). The control law(32) was used in the dynamic model for the cases whenε = 0 (totally rigid) and when ε 6= 0 (flexible). Theresults observed in the Subsection 4.2 indicates thatthe consideration of the flexible system is better thanthe rigid system. However, the deviations observed inthe first column of Table 1 can be improved by cho-osing larger values for K1 and K2 (parameters thatrepresent a proportional and derivative adjustment)or a larger value for N (i.e., more corrective actions).

Acknowledgment

The authors would like to thank to the CAPES (Co-ordenacao de Aperfeicoamento de Pessoal de Nıvel Su-perior), to the CNPq (Conselho Nacional de Desen-volvimento Cientıfico e Tecnologico) and to the FA-PESB (Fundacao de Amparo a Pesquisa do Estadoda Bahia) for the support given to this research.

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−3 −2 −1 0 1 2 3 4

−3

−2

−1

0

1

2

3

4

−0.2 −0.1 0 0.12.9

3

3.1

Referência2.34 cm/s4.68 cm/s7.02 cm/s9.36 cm/s

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y1 (m)

(a) Tracking trajectory for the WMR with a gradual in-crease in the velocity vn by using the control law (32) inthe model (9)-(11) for ε 6= 0.

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3

4

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Referência2.34 cm/s4.68 cm/s7.02 cm/s9.36 cm/s

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y1 (m)

(b) Tracking trajectory for the WMR with a gradual in-crease in the velocity vn by using the control law (32) inthe model (9)-(11) for ε = 0.

Figure 5: Tracking trajectory for the WMR with a gradual increase in the velocity vn.

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